An electric dipole has a fixed dipole moment P which makes angle … - Vidyadip

Question

An electric dipole has a fixed dipole moment $\vec P$ which makes angle $\theta $ with respect to $x-$axis. When subjected to an electric field $\overrightarrow {{E_1}} $ $=E$$\hat i$ it experiences a torque $\overrightarrow {{T_1}} $ =$\;\tau \hat k$ When subjected to another electric field $\overrightarrow {{E_2}} = \sqrt 3 {E_1}\hat j$ it experiences torque $\overrightarrow {{T_2}} $ = $ - \overrightarrow {{T_1}} $ The angle $\theta \;$ is.......$^o$

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Answer

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$\mathrm{T}=\mathrm{PE} \sin \theta$ Torque experienced by the dipole in an electric field,

$\quad \vec{T}=\vec{P} \times \vec{E}$

$\vec{p}=p \cos \theta \hat{i}+p \sin \theta \hat{j}$

$\vec{E}_{1}=\overrightarrow{E i}$

$\vec{T}_{1}=\vec{p} \times \vec{E}_{1}=(p \cos \theta \hat{i}+p \sin \theta \hat{j}) \times E(\hat{i})$

$\tau \hat{k}=p E \sin \theta(-\hat{k}).........(i)$

$\vec{E}_{2}=\sqrt{3} E_{1} \hat{j}$

$\left.\vec{T}_{2}=p \cos \theta \hat{i}+p \sin \theta \hat{j}\right) \times \sqrt{3} E_{1} \hat{j}$

$\tau \hat{k}=\sqrt{3} p E_{1} \cos \theta \hat{k}.........(ii)$

From eqns. ( $i$ ) and $(ii )$

$p E \sin \theta=\sqrt{3} p E \cos \theta$

$\tan \theta=\sqrt{3} \quad \therefore \quad \theta=60^{\circ}$

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